# Solving Recursions like this

How can i solve this equation? I am really stuck

$T(n) = T(n + 1) + T(n + 2) + 3n + 1$

$T(0)=2$

$T(1)=3$

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First you solve the homogeneous version (without the $3n+1$), using the characteristic equation. –  The Chaz 2.0 May 9 '13 at 12:43
A usual way to do that is to solve the homogeneous part. And then to find a particular solution by the method of undetermnined coefficients. Finally sum these two and determine the constants thanks to the initial conditions. –  1015 May 9 '13 at 12:43
Here is the method. Do not forget to upvote the answers if you benefit from them. –  Mhenni Benghorbal May 9 '13 at 12:44
–  Mhenni Benghorbal May 9 '13 at 13:23
@vadim123: Just scroll the page down till you reach the answer you want and then copy the address. –  Mhenni Benghorbal May 9 '13 at 13:36

Setting $S_n=T(n)+3n$, you'll obtain $$S_n=S_{n+1}+S_{n+2}-8.$$
Just set $F_n=T(n)+3n-8$ immediately and you obtain the Fibonacci recursion. –  Raskolnikov May 9 '13 at 12:46